Derived length of a Frobenius-like kernel

Date

2014-08-15

Author

Ercan, Gülin
Khukhro, Evgeny

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A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that FH/[F,F] is a Frobenius group with Frobenius kernel F/[F,F]. Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide vertical bar FH vertical bar. It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C-V(H) of the fixed-point subspace of H by g(m) = 3 + [log(2)(m + 1)]. It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most g(r), where r is the sectional rank of C-G(H). As an application, for a finite solvable group G admitting an automorphism phi of prime order coprime to vertical bar G vertical bar, a bound for the p-length of G is obtained in terms of the rank of a Hall p'-subgroup of C(phi). Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its. fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic). (C) 2014 Published by Elsevier Inc.

Subject Keywords

Frobenius-like group, Derived length, Fixed points

URI

https://hdl.handle.net/11511/35713

Journal

JOURNAL OF ALGEBRA

DOI

https://doi.org/10.1016/j.jalgebra.2014.04.025

Collections

Department of Mathematics, Article

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Action of a Frobenius-like group with kernel having central derived subgroup Ercan, Gülin (2016-09-01) A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that [F, h] = F for all nonidentity elements h is an element of H. Suppose that a finite group G admits a Frobenius-like group of auto-morphisms FH of coprime order with [F', H] = 1. In case where C-G( F) = 1 we prove that the groups G and C-G( H) have the same nilpotent length under certain additional assumptions.
NILPOTENT LENGTH OF A FINITE SOLVABLE GROUP WITH A FROBENIUS GROUP OF AUTOMORPHISMS Ercan, Gülin; Ogut, Elif (2014-01-01) We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C-CG(F) (h) = 1 for all nonidentity elements h is an element of H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

Citation Formats

G. Ercan and E. Khukhro, “Derived length of a Frobenius-like kernel,” JOURNAL OF ALGEBRA, pp. 179–188, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/35713.